Hopf Bifurcation and Stability Analysis of a Business Cycle Model With TimeDelayed Feedback

Hopf Bifurcation and Stability Analysis of a Business Cycle Model With Time-Delayed Feedback

  IJETT-book-cover  International Journal of Engineering Trends and Technology (IJETT)          
  
© 2018 by IJETT Journal
Volume-61 Number-3
Year of Publication : 2018
Authors : Lei Peng, Yanhui Zhai
DOI :  10.14445/22315381/IJETT-V61P223

Citation 

MLA Style: Lei Peng, Yanhui Zhai "Hopf Bifurcation and Stability Analysis of a Business Cycle Model With Time-Delayed Feedback" International Journal of Engineering Trends and Technology 61.3 (2018): 138-148.

APA Style:Lei Peng, Yanhui Zhai, (2018). Hopf Bifurcation and Stability Analysis of a Business Cycle Model With Time-Delayed Feedback. International Journal of Engineering Trends and Technology, 61(3), 138-148.

Abstract
In this paper, a business cycle model with time-delayed feedback is investigated. Firstly, we add a time-delayed feedback controller to the business cycle model and propose a new model. Secondly, the linear stability of the model and the local Hopf bifurcation are studied and we derive the conditions for the stability and the existence of Hopf bifurcation at the equilibrium of the system. Besides, the direction of Hopf bifurcation and the stability of bifurcation periodic solutions are studied by adopting the center manifold theorem and the normal form theory. At last, some numerical simulation results are presented to confirm that the controller can effectively increase the stability region of the business cycle model.

Reference
[1] C.H. Zhang, X.P. Yan, G.H. Cui. (2010). Hopf bifurcations in a predator–prey system with a discrete delay and a distributed delay. Nonlinear Analysis Real World Applications, 11(5), 4141-4153.
[2] J. Li, W. Xu, W. Xie, Z. Ren. (2008). Research on nonlinear stochastic dynamical price model. Chaos Solitons & Fractals, 37(5), 1391-1396.
[3] D. Tao, X. Liao, T. Huang. (2013). Dynamics of a congestion control model in a wireless access network. Nonlinear Analysis: Real World Applications, 14(1), 671-683.
[4] T. Puu, Irina Sushko. (2004). A business cycle model with cubic nonlinearity. Chaos, Solitons and Fractals, 19(3), 597-612.
[5] Chuirui Zhang, Junjie Wei. (2004). Stability and bifurcation analysis in a kind of business cycle model with delay. Chaos, Solitons and Fractals, 22(4), 883-896.
[6] X.D. Liu, W.L. Cai, J.J Lu, et al. (2015). Stability and Hopf bifurcation for a business cycle model with expectation and delay. Communications in Nonlinear Science & Numerical Simulation, 25(1-3), 149-161.
[7] M. Szydlowski, A. Krawiec. (2005). The stability problem in the Kaldor-Kalecki business cycle model. Chaos Solitons & Fractals. 25(2), 299-305.
[8] A. Kaddar, H. Talibi Alaoui. (2008). Hopf bifurcation analysis in a delayed Kaldor-kalecki model of business cycle. Nonlinear Analysis Modelling & Control, 13(4), 439-449.
[9] J. Yu, M. Peng. (2016). Stability and bifurcation analysis for the Kaldor-kalecki model with a discrete delay and distributed delay. Physica A Statistical Mechanics & Its Applications, 460, 66-75.
[10] Junhai Ma, Qin Gao. (2009). stability and Hopf bifurcations in a business cycle model with delay. Applied Mathematics and Computation, 215(2), 829-834.
[11] Jinchen Yu, Mingshu Peng, Caiyan Zhang. (2013). Hopf bifurcation of a business cycle model with time dealy. Journal of Beijing Jiaotong University, 37(3), 139-142.
[12] J. Hale. (1977). Theory of Functional Differential Equations, Springer.
[13] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan. (1981). Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge.
[14] D. Ding, J. Zhu, X.S. Luo. (2009). Delay induced Hopf bifurcation in a dual model of Internet congestion. Nonlinear Analysis: Real World Applications, 10(1), 2873-2883.
[15] Z.S. Cheng, J.D. Cao. (2014). Hybrid control of Hopf bifurcation in complex networks with delays. Neurocomputing, 131(131), 164-170.
[16] D. Fan, J. Wei. (2008). Hopf bifurcation analysis in a tri-neuron network with time delay, Nonlinear Analysis: Real World Applications, 9(1), 9-25.
[17] S. Guo, H. Zheng, and Q. Liu. (2010). Hopf bifurcation analysis for congestion control with heterogeneous delays. Nonlinear Analysis: Real World Applications,11(4), 3077-3090.
[18] Dawei Ding, Xuemei Qin, et.al. (2014). Hopf bifurcation control of congestion control model in a wireless access network. Neurocomputing, 144(1), 159-168.
[19] L.W. Liang, X.D. Wang, M. Peng. (2014). Hopf bifurcation analysis for a ratio-dependent predator-prey system with two delays and stage structure for the predator. Applied Mathematics and Computation, 231, 214-230.
[20] Y. Zhai, H. Bai, Y. Xiong, and X. Ma. (2013). Hopf bifurcation analysis for the modifed Rayleigh price model with time delay. Abstract and Applied Analysis, 2013(3), 432-445.
[21] Y.G. Zheng, Z.H. Wang. (2010). Stability and Hopf bifurcation of a class of TCP/AQM networks, Nonlinear Analysis: Real World Applications, 11(3), 1552-1559.
[22] Y. Chen, J. Liu. (2008). Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems. Applied Mathematics & Mechanics, 29(2), 199-206.

Keywords
Business cycle model,Time-delayed feedback, Stability, Hopf bifurcation, Numerical simulation.